TargetIIT

[im]http://codecogs.izyba.com/gif.latex?%5Cint%20%5Cfrac%7Bdx%7D%7B%281+%5Csqrt%7Bx%7D%29%5Csqrt%7Bx-x%5E%7B2%7D%7D%7D[/im]

edited: x = cos[p]2[/p] 2θ seems to work

It worked ,

thank you SOOOOO MUCHHH sir !!! [1]

 

[im]http://codecogs.izyba.com/gif.latex?%5Cint%20%5Cfrac%7Bdx%7D%7B%201+%20%5Csqrt%7Bx%7D.%5Csqrt%7Bx%7D.%5Csqrt%7B1-%20x%7D%7D[/im]

[im]http://codecogs.izyba.com/gif.latex?%5Csqrt%7Bx%7D%20%3D%20t[/im]


on differentiating

[im]http://codecogs.izyba.com/gif.latex?%5Cfrac%7B1%7D%7B2%5Csqrt%7Bx%7D%7D%20dx%20%3D%20dt[/im]

now integral becomes


[im]http://codecogs.izyba.com/gif.latex?2%20%5Cint%20%5Cfrac%7Bdt%7D%7B%28%201%20+%20t%20%29%20.%20%5Csqrt%7B1-%20t%5E2%20%7D%7D[/im]

now put ----[im]http://codecogs.izyba.com/gif.latex?1%20+%20t%20%3D%20%5Cfrac%7B1%7D%7Bz%7D%20%5CRightarrow%20t%20%3D%20%5Cfrac%7B1%7D%7Bz%7D%20-1%20%3D%20%5Cfrac%7B1-%20z%7D%7Bz%7D[/im]


[im]http://codecogs.izyba.com/gif.latex?dt%20%3D%20%5Cfrac%7B-1%7D%7Bz%5E2%20%7D[/im]


integral reduces to


[im]http://codecogs.izyba.com/gif.latex?-%202%5Cint%20%5Cfrac%7Bdz%20%7D%7B%5Csqrt%7B2z%20-%201%7D%7D%20%3D%20-%202%20%5Csqrt%7B2z%20-%201%7D[/im]

now put the  value of  z we get answer as

[im]http://codecogs.izyba.com/gif.latex?%5Cfrac%7B2%28%5Csqrt%7Bx%7D-1%29%7D%7B%5Csqrt%7B1%20-%20x%7D%7D%20+%20c[/im]



hope  this helps but hari sir 's  method is very  nice and ShOrT

yup ur ans is also correct [1], nice solution omkar!!